Permutations

A permutation is an ordered arrangement — first, second, third — where rearranging counts as different. Race podiums, passwords, and rankings are permutations. They're factorials, focused on choosing and ordering only some of the items.

Build the intuition

Ordered selection

Pick k items from n, and care about the order: the first slot has n choices, the second n−1, and so on for k slots. Multiply those k factors. For a gold-silver-bronze podium from 8 runners: 8 × 7 × 6 = 336 possible podiums. Order matters, so runner A-then-B differs from B-then-A.

P(n, k) = \frac{n!}{(n-k)!}

Reading the formula

The formula n!/(n−k)! looks fancy but just says 'multiply the top k factors of n!'. The (n−k)! in the denominator cancels the factors you don't use. When k = n (arrange everything), it reduces to n!/0! = n! — full permutations, the previous lesson. The formula is bookkeeping for 'one fewer choice each slot'.

Factorial growth, harnessed

Permutation counts grow nearly as fast as factorials — which is why brute-forcing an ordered arrangement is hopeless for even modest n. Ten people can line up in 10! ≈ 3.6 million orders; a deck of 52 cards has 52! ≈ 8 × 10⁶⁷ orderings, more than atoms in the galaxy. Every shuffle you make has almost certainly never existed before.

See it move

InteractiveThe counting lab

Stages2

Choices each3

2 stages, each with 3 choices → 3^2 = 9 outcomes. The fundamental counting principle: independent choices multiply.

Switch to 'Order matters' and fill the ordered slots: each slot offers one fewer choice than the last. The product is the permutation count.

A worked example

Count the race podiums

8 runners compete for gold, silver, and bronze. How many distinct podiums?
Ordered selection of 3 from 8:
$P(8, 3) = 8 \times 7 \times 6 = 336$
Or via the formula: 8!/(8−3)! = 8!/5! = 336 — the 5! cancels the unused factors.
336 podiums, because gold-A/silver-B differs from gold-B/silver-A. Order is the whole point.

Out in the world

Why shuffles are unique

Streaming services and card games rely on the astronomical count of 52! orderings: a well-shuffled deck is, with overwhelming probability, in an arrangement no deck in history has ever held. Permutation counting quantifies that 'practically impossible to repeat' — a fact that feels like magic but is just factorial growth.

Common confusion, cleared

“Permutations and combinations are interchangeable.”

Permutations count ordered arrangements; combinations count unordered selections. They differ by a factor of k! — often a huge gap. The first question is always: does reordering create a new outcome?

“The formula is something to memorize blindly.”

It's just 'n choices, then n−1, then n−2, for k slots'. If you remember the slot-filling story, you can reconstruct n!/(n−k)! every time.

Check yourself

PracticeQuick check

How many ways to award gold, silver, bronze among 6 athletes?

Recap

A permutation is an ordered selection — rearranging counts as new.
P(n, k) = n!/(n−k)! = the top k factors of n!.
Permutation counts grow factorially — astronomically fast.

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